The Centrality of Groups and Classes
M.G. Everett
School of Computing and Mathematical Sciences
University of Greenwich
Wellington Street
London SE 18 6PF
S.P. Borgatti
Organization Studies Dept.
Carroll Graduate School of Management
Boston College
Chestnut Hill, MA 02167
To appear in:
Journal of Mathematical Sociology, 1999.
Abstract
This paper extends the standard network centrality measures of degree, closeness and
betweenness to apply to groups and classes as well as individuals. The group centrality
measures will enable researchers to answer such questions as ‘how central is the engineering
department in the informal influence network of this company?’ or ‘among middle managers
in a given organization, which are more central, the men or the women?’ With these measures
we can also solve the inverse problem: given the network of ties among organization
members, how can we form a team that is maximally central? The measures are illustrated
using two classic network data sets. We also formalize a measure of group centrality
efficiency, which indicates the extent to which a group’s centrality is principally due to a
small subset of its members.
2
1 Introduction
Network analysts have used centrality as a basic tool for identifying key individuals in a
network since network studies began. It is an idea that has immediate appeal and as a
consequence is used in a large number of substantive applications across many disciplines.
However, it has one major restriction: with few exceptions, all published measures are intended
to apply to individual actors. It is a simple matter to think of areas of application that would
benefit from a formulation that applies to groups of actors rather than individuals. Are the
lawyers more central than the accountants in a given organization’s social network? Is one
particular ethnic minority more integrated into the community than another? To what extent are
particular groups or classes (women, the elderly, African-Americans, etc.) marginalized in
different networks? All of these questions could be answered to some extent by the application
of a centrality measure that applied to a set of individuals rather than a single individual.
In addition to a priori groups like the ones mentioned above, a group centrality measure could
also be applied to sets of individuals identified by cohesive subgroup techniques (e.g., cliques),
or by positional analysis techniques, such as structural equivalence or regular equivalence. For
example, if applied to cliques in very large networks, we could use group centrality to identify
which of many hundreds of cliques were the most important (in a well-specified sense) and
should be analyzed more fully.
Another application of a group centrality measure would be as a criterion for forming groups.
That is, we can write an algorithm to construct a set of groups, optionally mutually exclusive,
that have maximal group centrality. Or, less ambitiously, a manager wanting to put together a
team for a highly politically charged project might choose individuals who, in addition to having
the appropriate skills, would also maximize the team’s centrality.
2 General Principles
In order to develop a measure of group centrality we must first establish the criteria for success:
the features and properties that we would like such a measure to possess. Our first requirement
is that a group centrality measure be derived from existing individual measures. There are more
than enough measures of centrality already in the literature, and we do not intend to introduce
any more. Hence, what we introduce is a general method for applying existing measures to the
group context, rather than new conceptions of centrality. Our second requirement is that any
group measure be a proper generalization of the corresponding individual measure, such that
when applied to a group consisting of a single individual, the measure yields the same answer as
the individual version. An immediate consequence of this requirement is that we do not measure
group centrality by computing centrality on a network of relationships among groups. Instead,
the centrality of a group is computed directly from the network of relationships among
individuals. A side benefit of this approach is that there are no problems working with
overlapping groups, where one individual can belong to many groups.
An obvious approach to measuring group centrality would be to average or sum the individual
centrality scores of group members (possibly disregarding ties to other group members). This
approach has a number of problems. In a competitive situation, groups may form which seek to
gain an advantage by having high centrality scores. Clearly it would de disadvantageous for the
individual with the highest centrality to join with anyone else since the group centrality score
would almost certainly be lower. More generally if a group had an average centrality score and
an individual wanted to join them, unless the scores were equal, either the group should reject
2
the individual or the individual should reject the group. This problem would also prevent us
from achieving one of our goals, namely to allow us to use the measure as a criterion for
forming groups. Another problem with using an averaging method is that it takes no account
of the fact that actors may be central to (connected to) the same or different actors. For
example, suppose we have a group X with a certain group centrality score and two actors y
and z, where y is central to the same actors as the group X but z is central to a different set of
actors. If y and z have the same centrality score, then by the averaging method the groups
X+y and X+z will also have the same score -- but clearly the X+z group should have a better
score.
3 Group Centrality
In this paper we consider four measures of centrality: degree, closeness, betweenness, and flow
betweenness. For the sake of clarity of exposition, we shall assume that the data consist of a
connected, non-directed non-valued graph. However, the extension to non-symmetric and
valued data does not present any special problems.
Figure 1
Degree. We define group degree centrality as the number of non-group nodes that are
connected to group members. Multiple ties to the same node are counted only once. Hence, in
3
Figure 1, the centrality of the group consisting of nodes a and b is 6. We can normalize group
degree centrality by dividing the group degree by the number of non-group actors. Hence, the
normalized degree centrality of the group {a,b} is 1.0.
Figure 2
As an example of group centrality we shall look first at the primate data collected by Linda
Wolfe and given as a standard dataset in UCINET (Borgatti, Everett and Freeman 1992). The
data records 3 months of interactions amongst a group of 20 monkeys, where interactions were
defined as joint presence at the river. The dataset also contains information on the sex and age
of each animal. We shall consider six different groups. The first two groups will be formed by
sex; the remaining four will be formed by age. The purpose of dividing them by age is merely a
device to illustrate the techniques in this paper and we should emphasize that we have no
substantive reason for these groupings. The data is symmetric and valued and we have
4
dichotomized it by taking the presence of a tie to be more than 6 interactions over the time
period (see Figure 2).
Table 1. Individual Centrality Scores
Monkey
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Age
Group
14-16
10-13
10-13
7-9
7-9
14-16
4-5
10-13
7-9
7-9
14-16
10-13
14-16
4-5
7-9
10-13
7-9
4-5
14-16
4-5
Norm. Closene
Sex
Degree Degree ss
Male
4
21.05
142
Male
0
0
380
Male
13
68.42
133
Male
3
15.79
143
Male
2
10.53
144
Female
0
0
380
Female
3
15.79
143
Female
3
15.79
143
Female
1
5.26
145
Female
3
15.79
143
Female
2
10.53
144
Female
9
47.37
137
Female
6
31.58
140
Female
4
21.05
142
Female
6
31.58
140
Female
0
0
380
Female
3
15.79
143
Female
0
0
380
Female
0
0
380
Female
0
0
380
Norm.
Norm.
Norm. Flow
Flow
Closene Betwee Closene Betwee Betwee
ss
nness ss
nness nness
13.38
1
0.58
18
8.41
5
0
0
0
0
14.29
44.5
26.02
91
44.39
13.29
0
0
5
2.28
13.19
0
0
10
4.37
5
0
0
0
0
13.29
0
0
6
2.74
13.29
0.5
0.29
16
7.31
13.1
0
0
0
0
13.29
0
0
4
1.83
13.19
0
0
1
0.44
13.87
10.33
6.04
45
21.95
13.57
1.83
1.07
24
11.54
13.38
0
0
6
2.8
13.57
1.83
1.07
24
11.54
5
0
0
0
0
13.29
0
0
4
1.83
5
0
0
0
0
5
0
0
0
0
5
0
0
0
0
Table 1 gives the individual centralities for each monkey on four centrality measures, including
both normalized and un-normalized versions. Table 2 gives the group degree centrality and
normalized group degree centrality for the six groups.
Table 2. Group Degree Centrality
Group
1. Age 14-16
2. Age 10-13
3. Age 7-9
4. Age 4-5
5. Male
6. Female
Members
1 6 11 13 19
2 3 8 12 16
4 5 9 10 15 17
7 14 18 20
1-5
6-20
Group Degree
Centrality
8
11
5
5
10
4
5
Normalized Degree
Group Centrality
0.53
0.73
0.36
0.31
0.67
0.80
Among the age groups, the most central group is clearly the 10-13 year olds. This is the group
that contains monkey 3, who (as shown in Table 1) is highly central as an individual. The effect
of normalization is readily apparent in comparing groups 3 and 4: they have the same raw group
centrality score but group 3 is more central once the data have been normalized. The effect is
even more dramatic when we look at the male and female groups. Un-normalized, the males are
clearly more central than the females. But normalized, the situation is reversed. It is clearly
easier for larger groups to achieve higher normalized centrality scores than smaller groups
because they contain more individuals to connect with a smaller outside group. We shall return
to this point in the next section. Normalization has greater significance in group centrality than
in individual centrality. This is
because the differing sizes of groups mean that the
transformation is non-linear and hence the rank order of the normalized group centralities can be
quite different from the un-normalized ones.
Figure 3
As we have already mentioned, we can use group centrality to examine emergent groups
(revealed by standard network analysis procedures) as well as a priori classifications. Our
second empirical example uses the Bank Wiring Room data of Roethlisberger and Dickson
6
(1939), available in UCINET as well. In particular we examine the Games matrix (see Figure 3).
Isolates I3 and S2 were deleted before performing any analysis. A clique analysis of these data
finds 5 cliques with a considerable amount of overlap of the groups. Table 3 gives the group
degree and normalized group degree centrality for the cliques together with the members of the
groups.
Table 3. Group Degree Centrality of Bank Wiring Room Cliques
Clique
1. I1 W1 W2 W3 W4
2. W1 W2 W3 W4 S1
3. W1 W3 W4 W5 S1
4. W6 W7 W8 W9
5. W7 W8 W9 S4
Group Degree
Centrality
2
2
3
2
2
Normalized Group
Degree Centrality*
0.286
0.286
0.429
0.250
0.250
*Isolates removed prior to computing normalized scores.
Clearly, clique 3 has the highest group centrality score, but all the values are fairly similar. The
first three groups are all the same size, and therefore, among those three, the raw and normalized
scores are proportional to each other.
There is an important point to be considered when we use the concept of group degree
centrality on cohesive subgroups. To some extent, the notion of a cohesive subgroup includes
the idea of many links within the group, and few links to outsiders (Borgatti, Everett and
Shirey, 1991). Indeed, certain types of cohesive subsets (e.g., LS sets) are explicitly
constructed in such a way that they must have weak links to the rest of the network. Such
groups necessarily have low group degree centrality. Groups that have high degree centrality
are groups with highly porous or ambiguous boundaries.
Closeness. We can define group closeness as the sum of the distances from the group to all
7
vertices outside the group. As with individual closeness, this produces an inverse measure of
closeness as larger numbers indicate less centrality. This definition deliberately leaves
unspecified how distance from the group to an outside vertex is to be defined. This problem has
been well researched in the hierarchical clustering literature (Johnson, 1967) and we propose to
adopt their methods. Consider the set D of all distances from a single vertex to a set of vertices.
We can define the distance from the vertex to the set as either the maximum in D, the minimum
in D or the mean of values in D.1 For example, in Figure 2, the group consisting of {8,1,7} is
distance 1 from node 12 via the minimum method (because node 1 is just one link away from
12), distance 2 from 12 via the maximum method (because both nodes 7 and 8 are two links
from 12), and distance 1.67 via the mean method (because the average distance is (2+2+1)/3).
Of course, when the group consists of a single node, all of these distances are identical and the
group centrality is the same as individual centrality.
Following Freeman’s (1979) convention, we can normalize group closeness by dividing the
distance score into the number of non-group members, with the result that larger numbers
indicate greater centrality. Tables 4 and 5 give the group closeness for the primate and games
data using the same group numbering as before. In the primate data we have permanently
deleted the isolates 2,6,16,18, 19 and 20, and in the Games data we have deleted the two isolates
I3 and S2.
Table 4. Group closeness for the Primate data.
Group
1
Minimum
Mean
Maximum
Normalized
Normalized
Normalized
There are, of course, many other well-known variants, including the median method.
8
1. Age 14-16
2. Age 10-11
3. Age 7-9
4. Age 4-5
5. Male
6. Female
14
11
11
19
10
4
18
15
13.7
20.5
16
6.4
Minimum
0.79
1.00
0.73
0.63
1.00
1.00
20
21
15
22
20
7
Mean
0.61
0.73
0.58
0.59
0.63
0.63
Maximum
0.55
0.52
0.53
0.55
0.50
0.57
Table 5. Group closeness for the Games data.
Clique
Minimum
Mean
Maximum
1. I1 W1 W2 W3 W4
2. W1 W2 W3 W4 S1
3. W1 W3 W4 W5 S1
4. W6 W7 W8 W9
5. W7 W8 W9 S4
16
16
11
16
16
18.6
17.4
15.6
21.5
21.5
23
23
18
24
24
Normalized
Minimum
0.44
0.44
0.64
0.50
0.50
Normalized
Mean
0.38
0.40
0.45
0.37
0.37
Normalized
Maximum
0.30
0.30
0.39
0.33
0.33
As can be seen in Table 4, the minimum method does not provide much sensitivity: it is
relatively easy to attain the maximal value. In contrast, the maximum method is the most
stringent method, yielding the smallest value in all cases. The maximum and minimum methods
are similar in the sense that in both methods the distance of an individual to the group is defined
by the distance to a specific group member (either the closest or the furthest). In contrast, the
average method defines the distance in terms of all group members.
The choice of method for a given application will depend on the circumstances. If it is thought
that the group, once formed, acts as a single unit, then the minimum method is appropriate. In a
sense, the minimum method ignores internal structure (in particular, distances), and is therefore
almost equivalent to collapsing the group down to a single node whose ties are the union of the
ties to outsiders possessed by members. This may be appropriate when forming the group yields
a qualitatively different kind of agent, such as a corporation or other legal entity. Another
9
situation in which the minimum method might be appropriate is a communication network in
which cohesive groups have been identified consisting of individuals who have worked together
closely for many years, learning each other’s ways and developing the ability to communicate
with extraordinary efficiency. Even though individuals within the group are separated by a link
(i.e., they are still separate individuals), communication across internal links is virtually
instantaneous and complete, and so here again it is appropriate to ignore internal structure. In
contrast, when internal communication is not particularly good (or totally non-existent, as could
occur with classes defined on attributes), and it is important that all members of the group have
received all information, then the maximum method may be more appropriate. When the rules
of information transmission in the network suggest that a node transmits information to a
randomly chosen node in its neighborhood, the average method may be the best choice, as the
expected time-until-arrival of a message to the group will be a function of all the distance from
group members to all other nodes in the network.
Comparing the group closeness results with the group degree results we see a broad agreement
between the measures across both data sets. The only striking difference is in the centrality of
the male monkeys, where using the maximum method they are the least central of all the groups.
This is because they contain an individual placed slightly further away than the others all of
whom are very central. The minimum method will ignore him, the average method ameliorates
the effect, while the maximum method exposes the situation.
Betweenness. We now examine the third classic centrality measure, betweenness. The
properties of betweenness are radically different from those of degree and closeness and the
10
results are often correspondingly different. Let C be a subset of a graph with vertex set V. Let
gu,v be the number of geodesics connecting u to v and gu,v(C) be the number of geodesics
connecting u to v passing through C. Then the group betweenness centrality of C denoted by
CB(C) is given by
g (C)
 g
u ,v
u <v
u, v  C
u ,v
In other words the group betweenness centrality measure indicates the proportion of
geodesics connecting pairs of non-group members that pass through the group. One way to
compute this measure is as follows: (a) count the number of geodesics between every pair of
non-group members, yielding a node-by-node matrix of counts, (b) delete all ties involving
group members and redo the calculation, creating a new node-by-node matrix of counts, (c)
divide each cell in the new matrix by the corresponding cell in the first matrix, and (d) take
the sum of all these ratios.
As with individual betweenness centrality, we can normalize group betweenness by dividing
each value by the theoretical maximum. The theoretical maximum occurs for a group of a given
size when the result of identifying all the group vertices (i.e. shrinking them to a single vertex)
is a star with the group in the center. We therefore define the normalized group betweenness
centrality CB(C) as
CB(C) = 2 CB(C)/(V-C)(V-C-1)
Tables 6 and 7 give the group betweenness scores for the primate and games data.
11
Table 6. Group Betweenness for the Primate data
Group
1. Age 14-16
2. Age 10-11
3. Age 7-9
4. Age 4-5
5. Male
6. Female
Group
Betweenness
2.84
43.50
0.00
0.00
24.34
0.50
Normalized Group
Betweenness
0.03
0.41
0.00
0.00
0.23
0.05
Table 7. Group Betweenness for the Games data
Clique
1. I1 W1 W2 W3 W4
2. W1 W2 W3 W4 S1
3. W1 W3 W4 W5 S1
4. W6 W7 W8 W9
5. W7 W8 W9 S4
Betweenness
0.00
6.00
10.00
7.00
7.00
Normalized
Betweenness*
0.000
0.286
0.476
0.250
0.250
*Isolates removed
In examining the group betweenness results we notice that three of the groups have a value of
zero. It should be noted that only one of these groups consists of individuals all of whom have
individual betweenness centrality of zero – by joining a group, some individuals may “lose”
centrality. For the primate data we note the high scores achieved by group 2 and the male group.
If we look at individual betweenness for these data (see Table 1), we find that only six monkeys
-- 1,3,8,12,13 and 15 -- have non-zero betweenness. Of these, monkey 3 has by far the highest
score. Clearly groups that contain individuals with high individual centrality scores inherit some
of these scores, but not if the high scores were due to their connections with other group
12
members. Thus, there is a sense in which individuals can enhance their betweenness scores by
joining with individuals outside their own social circles.
Flow Betweenness. The fourth standard centrality measure is flow betweenness (Freeman,
Borgatti and White 1991). Individual flow betweenness can be extended to group flow
betweenness in the same way as ordinary betweenness. To calculate the group flow betweenness
of a group C, first calculate the sum of maximum flows in the network between all pairs of
vertices not in C. Call this quantity DEN. Then delete all ties involving nodes in C and repeat
the calculation, calling the result NUM. Finally, divide NUM by DEN. It should be noted that
group flow betweenness is quite a natural generalization of individual flow betweenness since
the computational algorithm we have presented is precisely the algorithm used for individual
flow betweenness (Borgatti, Everett and Freeman, 1992), where C contains just one vertex.
4 Efficiency
As we have already remarked, larger groups find it easier to get higher normalized group
centrality scores since they have to be central to (connected to) a smaller outside group. If we
were to look at the inverse of the problem we have been addressing so far, namely, given a
network, find a group with maximum group centrality, then we would quickly find it attractive
to look for large groups. Consider degree centrality: if we take any non-isolated vertex and place
all other vertices in a group, the group will have a normalized group degree centrality of one (an
example is provided by nodes a through g in Figure 1). The same would apply to the minimum
option in the group closeness measure. For betweenness we need to identify two non-adjacent
13
non-isolated vertices (such as g and d in Figure 1) and place all the other vertices in a group.
This group now has the maximum possible group betweenness score. Hence, by choosing large
groups, we can almost always maximize group centrality.
However, notice that in larger groups most individuals are not contributing directly to the group
centrality score. Their only contribution is to not be outside the group and therefore they do not
have to be included in the computation, which means they only contribute indirectly. The
presence of non-contributing members in a group suggests the concept of efficiency: the
proportion of actors within the group who directly contribute to the centrality of the group. Of
course, this conception of efficiency only makes sense for those centrality measure with the
property that adding actors to the group cannot decrease the centrality score. We now formalize
these concepts.
We say that a group centrality measure gpc is monotone if for every group C and subset K of C
gpc(K)  gpc(C). We assume here that the centrality measure assigns values in such a way that
higher numerical scores indicate more centrality (as in degree). If the measure assigns values
such that lower numerical scores indicate more centrality (as in closeness), then we reverse the
inequality.
The contribution of K for a monotone group centrality measure is the (un-
normalized) group centrality of K with respect to the actors outside C. As an illustration of this
concept consider degree centrality. The group centrality of C is the number of actors outside C
directly adjacent to the members of C. The contribution of K is the number of actors outside C
that are adjacent to a member of K. (We do not count actors who are in C and adjacent to K.) If
the contribution of K is as great as the group centrality of C then we say K makes a full
14
contribution and we call the members of K active members (implicitly we must have a
monotone group centrality measure). Let k be the size of the smallest subset K that makes a full
contribution to C. The efficiency e of a group C, with respect to a monotone group centrality
measure, is k divided by the size of C.
Degree, minimum closeness, betweenness and flow betweenness are all monotone group
centrality measures and so we can define efficiency when we use them. The average and
maximum closeness measures are not monotone hence efficiency is not defined for those
measures. For example, consider the clique {W1, W3, W4, W5, S1} of the Games data (see
Figure 3). The centrality of the group using the maximum method is 18. But the centrality of
actor W5 is 13, which is considerably better than the whole group.2 Clearly, we cannot measure
the contribution of W5 to the whole group since their individual contribution exceeds that of the
whole group, it follows that, in these circumstances, efficiency as we have developed it cannot
be applied. We note that any averaging method used to define group centrality would not be
monotone and consequently we would not be able to examine efficiency. Monotonicity is an
important property of group centrality and non-monotone methods should be used with extreme
caution. This is another reason that we do not advocate taking the average centrality score of a
group as a means of defining group centrality.
Table 8 gives the efficiency measure for each of the group centralities for the games data. Note
that betweenness efficiency for the first clique has not been computed since the betweenness of
the group is zero.
2
Recalling, of course, that unnormalized closeness is an inverse measure of centrality so that
a larger value indicates less centrality.
15
Table 8. Efficiency Scores for the Games Data
Clique
Degree
efficiency
0.20
0.20
0.40
0.25
0.25
1. I1 W1 W2 W3 W4
2. W1 W2 W3 W4 S1
3. W1 W3 W4 W5 S1
4. W6 W7 W8 W9
5. W7 W8 W9 S4
Closeness efficiency
(Minimum method)
0.20
0.20
0.40
0.25
0.25
Betweenness efficiency
NA*
0.80
0.80
0.25
0.25
*Group centrality is zero
For degree and closeness, the combination of several poorly connected actors and one or two
well-connected actors in a group means that the efficiency of the group must be low. However,
for betweenness centrality, the situation is quite different. The presence of an extremely central
member does not necessarily reduce group efficiency by making other nodes redundant. For
example, in Clique 3 of the Games data, actor W5 has enormous individual betweenness. But he
is in fact the one member of the clique that is expendable in the special sense that a subset of
Clique 3 that does not include him (namely, the subset K = {W1 W3 W4 S1}), achieves full
contribution as defined above.
5 Searching for Central Groups
As already mentioned, we could use the definition of group centrality as a basic tool to find
important subgroups within a given network. A manager who wishes to introduce some new
practices within an organization may wish to identify a well-connected subgroup that can
develop and champion the cause. This subgroup would need to have a very high group centrality
score. Yet to be effective, it would be desirable for the group to be as small as possible without
16
sacrificing centrality. We could therefore search for a minimal subgroup with maximal group
centrality.
For closeness or degree group centrality this would amount to finding the smallest group of
actors within the network such that every actor outside the group is adjacent to a member of the
group. This is a standard concept in graph theory, where the size of such a group is called the
domination number of a graph (Carre 1979). We may not have such a stringent requirement and
may only demand a group centrality greater than a certain value. Alternatively we may wish to
find a group with a fixed size which has the maximum group centrality. Or we may wish to find
a set of (possibly mutually exclusive) groups that, overall, maximize centrality. Any of these
searches amount to searching all subgroups for the group or groups which best fit the required
conditions. These therefore fall into the category of combinatorial optimization problems and
can be solved by any of the standard heuristic methods such as a genetic algorithm (Holland
1975).
We can perform such a search on the Games data. The domination number for the Games graph
is 2, and this can be achieved with a number of different pairs, including {W1, W7}, {W1,
W8}, and {W1, W9}.3 As we have already noted, the maximal betweenness occurs when the
group forms the center of a star. This can always be achieved by a group smaller than the whole
group, provided we do not have a complete graph. In the Games data, the smallest group with
maximal betweenness is surprisingly large (size 8), and there are three of these. One example is
{I1,S1,W1,W3,W4,W7,W8,W9}. This example is interesting because it does not contain the
3
The fact that W1 occurs in all of these is interesting and suggestive of a new measure of
individual centrality. But this is beyond the scope of the present paper.
17
highly central actor W5, although there is an alternative group of the same size which does
contain him.
6 Alternative Group Centrality Approach: The Reduced Model Approach
The methods and principles of the paper represent just one of a number of approaches that could
be taken to defining group centrality. In this section we briefly outline an alternative and discuss
some associated problems and advantages of this approach.
An intuitively obvious approach is to replace all the members of a group by a single “super”
vertex whose neighborhood is the union of the neighborhoods of all group members. That is,
there is a connection from the new vertex to another vertex if there was at least one actor in the
group who had that connection. If G was the original graph, then we call the new graph the
group reduced graph denoted by G*. If G had n actors and a group C had c actors then G*
contains n-c+1 actors. We could then submit G* to any standard individual centrality routine to
obtain the centrality measure for the group. This procedure would give exactly the same results
for degree centrality as we obtained using our group degree centrality measure. If we apply
closeness centrality to G* then we obtain the same results as the group closeness measure based
on the minimum method. We cannot, however, reproduce the mean and maximum closeness
methods using this technique.
18
< Figure 4 >
Differences also occur when we apply the betweenness centrality measure. The reason is that
the internal structure of the group has no effect on the reduced model, but does affect the group
betweenness measure we defined earlier. In the reduced model, connections internal to the
group do not exist, so geodesics passing through the group can be shorter. An example is
shown in Figure 4. Using our original group betweenness measure, the centrality of the group
{b, c} is zero, as the internal link increases the length of the path from a to d passing through the
group to 3 links, which is longer than the geodesic distance of 2.
The fact that, in the reduced model, the internal cohesion of a group does not affect its
centrality, is an important conceptual advantage of the reduced model. There are practical
benefits as well. In the earlier model, if the individual betweenness of each member of a group
is zero, the centrality of the group must be zero as well, much like the averaging method.
Therefore, for completely peripheral individuals, there is no strategic benefit to forming a group.
In contrast, using the reduced model approach, the centrality of the group in Figure 4 is 0.5, as
there are now two geodesics from a to d, and one of them passes through the group. This
approach therefore lends itself to a more strategic view of group formation, perhaps based on a
rational actor model. For example, a peripheral actor in a communication network might seek to
19
increase their betweenness centrality by partnering with other peripheral actors, such that the
combination has high centrality. This assumes, of course, that once actors are teamed,
communication among members is complete and instantaneous.
Tables 9 and 10 give the results of the reduction method on our two data sets using betweenness
centrality. For the Games data, the betweenness scores were exactly the same as using the
original group betweenness definition. This is not the case for the primate data. We can see in
Table 10 that all the group centrality scores have increased and two of the groups that had a zero
score now have a positive betweenness value.
Table 9. Reduced-graph group centrality for the Games data
Clique
Betweenness
1. I1 W1 W2 W3 W4
2. W1 W2 W3 W4 S1
3. W1 W3 W4 W5 S1
4. W6 W7 W8 W9
5. W7 W8 W9 S4
0.00
6.00
10.00
7.00
7.00
Normalized
Betweenness*
0.000
0.286
0.476
0.250
0.250
*Isolates removed
Table 10. Reduced-graph group centrality for the Primate data.
Group
Betweenness
1. Age 14-16
2. Age 10-11
3. Age 7-9
4. Age 4-5
5. Male
6. Female
6.33
43.33
1.33
0.67
24.33
1.5
Normalized
Betweenness
0.06
0.41
0.01
0.01
0.23
0.02
Another advantage of the reduced graph approach is that it allows us to construct group versions
20
of any centrality measure, including ones that are difficult to generalize along the lines presented
earlier in this paper. An example of such a measure would be eigenvector centrality (Bonacich,
1972). Eigenvector centrality is defined as the principal eigenvector of the adjacency matrix for
the graph. As such, it is virtually impossible to generalize along the lines presented earlier, but
easy to handle via the reduced graph approach.4
The basic criticism of the reduced graph method is that the removal of the internal links in a
group is hard to justify substantively. If in the network in general the links represent communication channels among actors, and if longer paths take longer to traverse (or involve decay
of usable information), how is it that, by forming a group, these basic laws of propagation are
suspended within the group? It is hard to imagine what substantive processes might correspond
to these algorithmic procedures.
7 Conclusion
We have proposed some general principles that have allowed us to develop centrality measures
for groups and classes in networks. These have been illustrated on two empirical datasets, using
both a priori classes of nodes such as age and sex and empirically derived groups such as
cliques. We believe the potential for group centrality measures as independent variables is
enormous. For example, in the team effectiveness literature, researchers have used a number of
internal team composition variables to predict performance (Katz 1982; Nelson and Winter
1982). More recently (Ancona 1990; Geletkanycz and Hambrick 1997), researchers have
4
Of course, radically different approaches, such as proposed by Bonacich (1991), are also
possible.
21
suggested that maintaining strong ties with people outside the team is an important determinant
of team success. However, measures of the group’s connections with the rest of the organization
have not been available. We believe that our measures fill this gap and that a fruitful line of
future research will use group centrality measures to predict the performance of teams in
organizations.
Another recent stream of research has focussed on strategic networking, particularly at the
individual level (Burt 1992; Baker 1994). For example, Burt (1992) considers how to evaluate
potential partners so as to maximize the reach of an actor’s ego network. Computationally, the
problem is equivalent to finding a maximally central subset of actors that include ego. An
interesting line of research would be to discover the conditions under which individuals’
sociometric choices do or do not accord with strategic principles (i.e., maximizing ego network
centrality). For example, we might expect that business contacts conform to strategic principles
while friendship contacts do not. Conversely, from a consulting or prescriptive point of view,
group centrality measures may provide a guide for selecting partners, either among persons or
corporations.
Group centrality measures may also provide an avenue for defining or operationalizing
substantive concepts. For example, the notion of a core/periphery structure is prevalent in many
literatures (notably the study of elites and in world systems theory). But no formal definition of
a core (or periphery) has been proposed. Given the existence of group centrality measures, it
would be quite natural to define the core of a network as that subset of actors whose group
centrality is greater than all others. Another example of operationalizing fundamental constructs
22
is provided by the notion of ‘social capital’. Social capital has come to mean many very
disparate things, but one of them is surely the set of benefits that accrue from the set of ties that
an individual or group possesses with others in the network. Group centrality measures clearly
provide a way to operationalize this aspect of the concept of social capital.
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